A225768 Least k > 0 such that k^6 + n is prime, or 0 if k^6 + n is never prime.

0, 1, 1, 2, 1, 18, 1, 2, 0, 2, 1, 54, 1, 28, 3, 2, 1, 18, 1, 2, 399, 26, 1, 6, 5, 2, 21, 0, 1, 288, 1, 4, 3, 2, 105, 6, 1, 2, 33, 2, 1, 546, 1, 2, 3, 2, 1, 6, 35, 2, 51, 20, 1, 12, 5, 28, 9, 4, 1, 18, 1, 4, 63, 2, 0, 18, 1, 2, 3, 28, 1, 6, 1, 2, 15, 2, 35, 24, 1, 12, 3, 4, 1, 42, 115, 2, 111, 2, 1, 18, 91, 6, 3, 2, 3, 6, 1, 28, 3, 2

Offset: 0

Author

M. F. Hasler, Jul 25 2013

Keywords

• nonn

Comments

Motivated by the "particularly poor polynomial" n^6+1091 (composite for n=1,...,3905) mentioned on Weisstein's page about prime generating polynomials.

We have a(n) = 0 if n is a cube n = g^3 with g > 1 because then k^6 + g^3 = (k^2 + g)*(k^4 - k^2*g + g^2), which can be prime only when n = g = 1. - T. D. Noe, Nov 18 2013

By the theorem of Brillhart, Filaseta and Odlyzko (see link), if a(n) > n > 1 then x^6 + n must be irreducible. If x^6 + n is irreducible, the Bunyakovsky conjecture implies a(n) is finite. - Robert Israel, Apr 25 2016

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• J. Brillhart, M. Filaseta, and A. Odlyzko, On an irreducibility theorem of A. Cohn, Canadian Journal of Mathematics 33(1981), 1055-1059.
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Crossrefs

See A085099, A225765, A225767, A225769, A225770 for the k^2, k^3, ..., k^8 analogs.

Programs

Maple

f:= proc(n) local exact, x,k,F,nf,F1,C;
    iroot(n,3,exact);
    if exact and n > 1 then return 0 fi;
    if irreduc(x^6+n) then
       for k from 1+(n mod 2) by 2 do if isprime(k^6+n) then return k fi od
    else
       F:= factors(x^6+n)[2]; #
       F1:= map(t -> t[1],F);
       nf:= nops(F);
       C:= map(t -> op(map(rhs@op,{isolve(t^2-1)})),F1);
       for k in sort(convert(select(type,C,positive),list)) do
         if isprime(k^6+n) then return k fi
       od:
       0
    fi
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016

Mathematica

{0, 1}~Join~Table[If[IrreduciblePolynomialQ[x^6 + n], SelectFirst[Range[1 + Mod[n, 2], 10^3, 2], PrimeQ[#^6 + n] &], 0], {n, 2, 120}] (* Michael De Vlieger, Apr 25 2016, Version 10 *)

PARI

{(a,b=6)->#factor(x^b+a)~==1 & for(n=1, 9e9, ispseudoprime(n^b+a)&return(n)); a==1&return(1);print1("/*"a":", factor(x^b+a)"*/")} /* For illustrative purpose only. The polynomial x^6+a is factored to avoid an infinite loop when it is composite. But there could be x such that this is prime, when all factors but one are 1 (not for exponent b=6, but, e.g., x=4 for exponent b=4), see A225766. */